Draw the variation table of a trigonometric function from a graph - High school Precalculus Exercises

Make conjectures about the variations of the following functions.

This function is increasing at every interval with the form \left[ k\pi;k\pi+\dfrac{\pi}{4} \right] and decreasing at every interval with the form \left[k\pi+\dfrac{\pi}{4};\left(k+1\right)\pi\right], for any k\in \mathbb{Z}.

This function is decreasing at every interval with the form \left[ k\pi;k\pi+\dfrac{\pi}{4} \right] and increasing at every interval with the form \left[k\pi+\dfrac{\pi}{4};\left(k+1\right)\pi\right], for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left[ k\pi;k\pi+\dfrac{\pi}{2} \right] and decreasing at every interval with the form \left[k\pi+\dfrac{\pi}{2};\left(k+1\right)\pi\right], for any k\in \mathbb{Z}.

This function is decreasing at every interval with the form \left[ k\pi;k\pi+\dfrac{\pi}{2} \right] and increasing at every interval with the form \left[k\pi+\dfrac{\pi}{2};\left(k+1\right)\pi\right], for any k\in \mathbb{Z}.

From left to right, a function is increasing if its graph is going "up" and is decreasing if its graph is going "down".

In the problem, the function is decreasing at every interval with the form \left[ k\pi;k\pi+\dfrac{\pi}{2} \right] and increasing at every interval with the form \left[k\pi+\dfrac{\pi}{2};\left(k+1\right)\pi\right], for any k\in \mathbb{Z}.

This function is decreasing at every interval with the form \left[ \dfrac{\pi}{4}+\left(2k-1\right)\pi,\dfrac{\pi}{4}+2k\pi\right] and increasing at every interval with the form \left[ \dfrac{\pi}{4}+2k\pi,\dfrac{\pi}{4}+\left(2k+1\right)\pi\right], for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left[ \dfrac{\pi}{4}+\left(2k-1\right)\pi,\dfrac{\pi}{4}+2k\pi\right] and decreasing at every interval with the form \left[ \dfrac{\pi}{4}+2k\pi,\dfrac{\pi}{4}+\left(2k+1\right)\pi\right], for any k\in \mathbb{Z}.

From right to left, a function is increasing if its graph is going "up" and is decreasing if its graph is going "down".

In the problem, the function is increasing at every interval with the form \left[ \dfrac{\pi}{4}+\left(2k-1\right)\pi,\dfrac{\pi}{4}+2k\pi\right] and decreasing at every interval with the form \left[ \dfrac{\pi}{4}+2k\pi,\dfrac{\pi}{4}+\left(2k+1\right)\pi\right], for any k\in \mathbb{Z}.

This function is decreasing at every interval with the form \left[ \dfrac{3\pi}{4}+2k\pi,\dfrac{3\pi}{4}+\left(2k+1\right)\pi\right] and increasing at every interval with the form \left[ -\dfrac{\pi}{4}+2k\pi,\dfrac{3\pi}{4}+2k\pi\right], for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left[ \dfrac{3\pi}{4}+2k\pi,\dfrac{3\pi}{4}+\left(2k+1\right)\pi\right] and decreasing at every interval with the form \left[ -\dfrac{\pi}{4}+2k\pi,\dfrac{3\pi}{4}+2k\pi\right], for any k\in \mathbb{Z}.

From left to right, a function is increasing if its graph is going "up" and is decreasing if its graph is going "down".

In the problem, the function is increasing at every interval with the form \left[ \dfrac{3\pi}{4}+2k\pi,\dfrac{3\pi}{4}+\left(2k+1\right)\pi\right] and decreasing at every interval with the form \left[ -\dfrac{\pi}{4}+2k\pi,\dfrac{3\pi}{4}+2k\pi\right], for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left( k\pi;k\pi+\dfrac{\pi}{4} \right), for any k\in \mathbb{Z}.

This function is decreasing at every interval with the form \left( k\pi;k\pi+\dfrac{\pi}{4} \right) and increasing at every interval with the form \left(k\pi+\dfrac{\pi}{4};\left(k+1\right)\pi\right), for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left( -\dfrac{\pi}{4}+k\pi,\dfrac{\pi}{4}+k\pi\right) and decreasing at every interval with the form \left( \dfrac{\pi}{4}+k\pi,\dfrac{\pi}{4}+\left(k+1\right)\pi\right), for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left( \dfrac{\pi}{4}+k\pi,\dfrac{\pi}{4}+\left(k+1\right)\pi\right), for any k\in \mathbb{Z}.

From left to right, a function is increasing if its graph is going "up" and is decreasing if its graph is going "down".

In the problem, the function is increasing at every interval with the form \left( \dfrac{\pi}{4}+k\pi,\dfrac{\pi}{4}+\left(k+1\right)\pi\right) and is not defined for x = \dfrac{\pi}{4} +k\pi, for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left[ k\pi,\dfrac{\pi}{2}+k\pi\right] and decreasing at every interval with the form \left[ \dfrac{\pi}{2}+k\pi,\left(k+1\right)\pi\right], for any k\in \mathbb{Z}.

This function is decreasing at every interval with the form \left[ k\pi,\dfrac{\pi}{2}+k\pi\right] and increasing at every interval with the form \left[ \dfrac{\pi}{2}+k\pi,\left(k+1\right)\pi\right], for any k\in \mathbb{Z}.

From left to right, a function is increasing if its graph is going "up" and is decreasing if its graph is going "down".

In this problem, the function is increasing at every interval with the form \left[ k\pi,\dfrac{\pi}{2}+k\pi\right] and decreasing at every interval with the form \left[ \dfrac{\pi}{2}+k\pi,\left(k+1\right)\pi\right], for any k\in \mathbb{Z}.

This function is decreasing at every interval with the form \left[ -\dfrac{\pi}{4}+k\pi,\dfrac{\pi}{4}+k\pi\right] and increasing at every interval with the form \left[ \dfrac{\pi}{4}+k\pi,\dfrac{3\pi}{4}+k\pi\right], for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left[ -\dfrac{\pi}{4}+k\pi,\dfrac{\pi}{4}+k\pi\right] and decreasing at every interval with the form \left[ \dfrac{\pi}{4}+k\pi,\dfrac{3\pi}{4}+k\pi\right], for any k\in \mathbb{Z}.

From left to right, a function is increasing if its graph is going "up" and is decreasing if its graph is going "down".

In this problem, the function is decreasing at every interval with the form \left[ -\dfrac{\pi}{4}+k\pi,\dfrac{\pi}{4}+k\pi\right] and increasing at every interval with the form \left[ \dfrac{\pi}{4}+k\pi,\dfrac{3\pi}{4}+k\pi\right], for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left( \dfrac{\pi}{4}+k\dfrac{\pi}{2},\left(k+1\right)\dfrac{\pi}{2}\right), for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left( \dfrac{\pi}{4}+k\dfrac{\pi}{2},\left(k+1\right)\dfrac{\pi}{2}\right) and decreasing at every interval with the form \left( k\dfrac{\pi}{2},\dfrac{\pi}{4}+k\dfrac{\pi}{2}\right), for any k\in \mathbb{Z}.

This function is increasing at every interval with the form \left( k\dfrac{\pi}{2};\left(k+1\right)\dfrac{\pi}{2}\right), for any k\in \mathbb{Z}.

From left to right, a function is increasing if its graph is going "up" and is decreasing if its graph is going "down".

In the problem, the function is increasing at every interval with the form \left( k\dfrac{\pi}{2};\left(k+1\right)\dfrac{\pi}{2}\right) and is not defined for x = k\dfrac{\pi}{2}, for any k\in \mathbb{Z}.